Modular $A_4$ invariance and neutrino mixing

We study the phenomenological implications of the modular symmetry $\Gamma(3) \simeq A_4$ of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus $\tau$, which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with $A_4$ symmetry. We classify our neutrino models along with the type I seesaw model, the Weinberg operator model and the Dirac neutrino model. In the normal hierarchy of neutrino masses, the seesaw model is available by taking account of recent experimental data of neutrino oscillations and the cosmological bound of sum of neutrino masses. The predicted $\sin^2\theta_{23}$ is restricted to be larger than $0.54$ and $\delta_{CP}=\pm (50^{\circ}\mbox{--}180^{\circ})$. Since the correlation of $\sin^2\theta_{23}$ and $\delta_{CP}$ is sharp, the prediction is testable in the future. It is remarkable that the effective mass $m_{ee}$ of the neutrinoless double beta decay is around $22$\,meV while the sum of neutrino masses is predicted to be $145$\,meV. On the other hand, for the inverted hierarchy of neutrino masses, only the Dirac neutrino model is consistent with the experimental data.


Introduction
In spite of the remarkable success of the standard model (SM), the origin of the flavors of quarks and leptons is still unknown. The recent developments of the neutrino oscillation experiments provide us important clues to investigate the flavor physics. Indeed, the neutrino oscillation experiments have determined two neutrino mass squared differences and three neutrino mixing angles precisely. In particular, the recent data of both T2K [1,2] and NOνA [3,4] imply that the atmospheric neutrino mixing angle θ 23 is expected to be near the maximal angle 45 • . The closer the observed θ 23 is to the maximal mixing, the more likely some flavor symmetry is to exist behind it. In addition to the precise measurements of neutrino mixing angles, T2K and NOνA strongly indicate CP violation in the neutrino oscillation [2,4]. Thus, we are in the era to develop the flavor theory of leptons with facing both the flavor mixing angles and CP violating phase.
One of interesting approaches is to impose non-Abelian discrete symmetries for flavors. Many models have been proposed by using S 3 , A 4 , S 4 , A 5 and other groups with lager orders [5][6][7][8][9]. In particular, A 4 flavor models are attractive because A 4 is the minimal group which has a triplet as its irreducible representation and enable us to explain three families of quarks and leptons naturally [10][11][12][13][14][15]. However, variety of models is so wide that it is difficult to obtain clear clues of flavor symmetry. Indeed, symmetry breakings are required to reproduce realistic mixing angles [16]. The effective Lagrangian of a typical flavor model is given by introducing the gauge singlet scalars which are so-called flavons. Those vacuum expectation values (VEVs) determine the flavor structure of quarks and leptons. Finally, the breaking sector of flavor symmetry typically produces many unknown parameters.
It is interesting that the modular group includes S 3 , A 4 , S 4 , and A 5 as its finite subgroups, Γ(N ). However, there is a difference between the modular symmetry and the usual flavor symmetry. Yukawa couplings are written as modular forms, functions of the modulus τ , and transform non-trivially under the modular symmetry as well as fields. On the other hand, Yukawa couplings are invariants in the usual flavor symmetries. In this aspect, an attractive ansatz was proposed by taking Γ(3) A 4 in Ref. [28] where Yukawa couplings are A 4 triplets of modular forms, and both left-handed leptons and right-handed neutrinos are A 4 triplets while right-handed charged leptons are A 4 singlets. Along with this work, Γ(2) S 3 [29] and Γ(4) S 4 [30] have been discussed as well as the numerical works [31]. These are bottom-up approaches for model building, but not a top-down approach from explicit string models. However, these approaches would make a bridge between neutrino physics and underlying theory such as superstring theory from the viewpoint of flavor symmetries.
In this paper, we present a comprehensive study of Γ(3) A 4 numerically by taking account of the recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by the expectation value of the modulus τ , which is the only source of modular invariance breaking. However, there are freedoms for the assignments of irreducible representations and modular weights to leptons. We study neutrino mass matrices for three classified models: the type I seesaw model, the Weinberg operator model, and the Dirac neutrino model. In order to build models with minimal number of parameters, we introduce no flavons.
The paper is organized as follows. In section 2, we give a brief review on modular symmetry. In section 3, we present the mass matrices for neutrinos and charged leptons in our models. In section 4, we present the numerical results of our models. Section 5 is devoted to a summary. Appendix A shows the relevant multiplication rules of A 4 . In Appendix B, we show how to determine the coupling coefficients of the charged lepton sector. Appendix C shows the lepton mixing matrix and the relevant measures which are used in this work for the case of Majorana neutrinos.

Modular symmetry
In this section, we give a brief review on the modular symmetry on the torus and its low-energy effective field theory.
The torus compactification is the simplest compactification. For example, the two-dimensional torus T 2 can be constructed as division of R 2 by a two-dimensional lattice Λ, i.e. T 2 = R 2 /Λ. Here, we use the complex coordinate on R 2 with the lattice spanned by two lattice vectors, α 1 = 2πR and α 2 = 2πRτ ; where R is real and τ is a complex modulus parameter. However, there is some ambiguity in choice of the basis vectors. The same lattice can be spanned by the following basis vectors, where a, b, c, d are integer with satisfying ad − bc = 1. That is the SL(2, Z) transformation. Under the above transformation, the modulus parameter transforms as and this modular transformation is generated by S and T , They satisfy the following algebraic relations, If we impose T N = I furthermore, we obtain finite subgroups Γ(N ). Γ(N ) with N = 2, 3, 4, 5 are isomorphic to S 3 , A 4 , S 4 and A 5 , respectively [32]. Indeed, Γ(N ) is a quotient of the modular group by the so-called congruence subgroupΓ(N ). Holomorphic functions which transform as under the modular transformation Eq.(2) are called modular forms of weight k.
Superstring theory on the torus T 2 or orbifold T 2 /Z N has the modular symmetry. Its lowenergy effective field theory is described in terms of supergravity theory, and string-derived supergravity theory has also the modular symmetry. Under the modular transformation Eq.(2), chiral superfields φ (I) transform as [33], where −k I is the so-called modular weight and ρ (I) (γ) denotes a unitary representation matrix of γ ∈ Γ(N ). The kinetic terms of their scalar components are written by which is invariant under the modular transformation. Here, we use the convention that the superfield and its scalar component are denoted by the same letter. Also, the superpotential should be invariant under the modular symmetry. That is, the superpotential should have vanishing modular weight in global supersymmetric models, while the superpotential in supergravity should be invariant under the modular symmetry up to the Kähler transformation. In the following sections, we study global supersymmetric models, e.g. minimal supersymmetric standard model (MSSM) and its extension with right-handed neutrinos. Thus, the superpotential has vanishing modular weight. However, note that Yukawa couplings as well as higher order couplings depend on modulus, and they can have non-vanishing modular weights. The breaking scale of supersymmetry can be between O(1)TeV and the compactification scale. The modular symmetry is broken by the vacuum expectation value of τ , i.e. at the compactification scale, which is of order of the Planck scale or slightly lower scale. The Dedekind eta-function η(τ ) is one of famous modular forms, which is written by where q = e 2πiτ and η(τ ) 24 is a modular form of weight 12. By use of η(τ ) and its derivative, A 4 triplet modular forms (Y 1 , Y 2 , Y 3 ) of modular weight 2 are written by [28], where ω = e 2πi/3 . The overall coefficient in Eq.(10) is one choice and cannot be determined essentially.

Models with modular symmetry
Let us consider a modular invariant flavor model with the A 4 symmetry for leptons. At first, we discuss the type I seesaw model where neutrinos are Majorana particles. There are freedoms for the assignments of irreducible representations and modular weights to leptons. We suppose that three left-handed lepton doublets are compiled in a triplet of A 4 . The three right-handed neutrinos are also of a triplet of A 4 . On the other hand, the Higgs doublets are supposed to be singlets of A 4 . The generic assignments of representations and modular weights to the MSSM fields and right-handed neutrino superfields are presented in Table 1. In order to build a model with minimal number of parameters, we introduce no flavons. For the charged leptons, we assign three right-handed charged leptons for three different singlets of A 4 , (1, 1 , 1 ). Therefore, there are three independent couplings in the superpotential of the charged lepton sector. Those coupling constants can be adjusted to the observed charged lepton masses. Since there are three singlets in the A 4 group, there are six cases for the assignment of three right-handed charged leptons. However, the freedom of these assignments for righthanded charged leptons do not affect the results for lepton mixing angles.
It may be helpful to comment that if the right-handed charged leptons are of a A 4 triplet, we cannot reproduce the well known charged lepton mass hierarchy 1 :  (2), A 4 , and the modular weight (−k I for fields and k for coupling Y ) in the type I seesaw model. The right-handed charged leptons are assigned three A 4 singlets, respectively. Values of −k I in the parentheses are alternative assignments of the modular weight.
The modular invariant mass terms of the leptons are given as the following superpotentials: where sums of the modular weights vanish. The parameters α, β, γ, g, and Λ are constant coefficients. The functions Y i (τ ) are A 4 triplet modular forms and they consist of the modulus parameter τ : where the q-expansion of Y i (τ ) is used. The Y i (τ ) satisfy the constraint [28]: Since the dimension of the space of modular forms of weight 2 for Γ(3) A 4 is 3 (see, e.g. [28,34]), all Y 's in Eqs.(11)-(13) are the same modular forms.
There is an alternative assignment of the modular weight for the left-handed lepton and the right-handed charged leptons as presented in parentheses of Table 1 [31]. For the alternative assignment, the modular invariant superpotential w D is given with constant parameters without the modular coupling Y as: Next, we discuss the case where neutrino masses originate from the Weinberg operator. We have the unique possibility of the superpotential where both modular weights of L and right-handed charged leptons are −1 as shown in Table 1.
There is another possibility for neutrinos, that is, neutrinos are Dirac particles. In this case, the neutrino mass matrix is derived only from w D in Eq.(12).

Charged lepton mass matrix
Let us consider an assignment of A 4 for the right-handed charged leptons as (e R , µ R , τ R ) = (1, 1 , 1 ) in Table 1. By using the decomposition rule of a A 4 tensor product in Appendix A, we obtain the mass matrix of charged leptons as follows 1 : The coefficients α, β, and γ are taken to be real positive by rephasing right-handed charged lepton fields without loss of generality. Those parameters can be written in terms of the modulus parameter τ and the charged lepton masses as seen in Appendix B.

Neutrino mass matrix
Since the tensor product of 3 ⊗ 3 is decomposed into a symmetric triplet and an antisymmetric triplet as seen in Appendix A, the superpotential of the Dirac neutrino mass in Eq. (12) is expressed with additional two parameters g 1 and g 2 as: The Dirac neutrino mass matrix is given as For the alternative case in Eq. (16), the superpotential of the Dirac neutrino is written as: The Dirac neutrino mass matrix is simply given as On the other hand, since the Majorana neutrino mass terms are symmetric, the superpotential in Eq. (13) is expressed simply as Then, the right-handed Majorana neutrino mass matrix is given as Finally, the effective neutrino mass matrix is obtained by the type I seesaw as follows: For the case where neutrino masses originate from the Weinberg operator, the superpotential in Eq.(17) is written as:

Models
Mass Matrices The Majorana neutrino mass matrix is given as follows:

I (a) : Seesaw
This matrix is the same one as in Eq.(24) apart from the normalization because both left-handed neutrinos and the right-handed neutrinos are the triplet of A 4 . For the case where the neutrino is the Dirac particle, we use the mass matrix in Eq. (20). It is important to address the transformation needed to put kinetic terms of matter superfields in the canonical form because kinetic terms are given in Eq. (8). The canonical form is realized by the overall normalization of the lepton mass matrices, which shifts our parameters such as where K φ denotes a coefficient of the kinetic term of Eq. (8). Hereafter, we rewrite α, β, γ, g i , and Λ for α , β , γ , g i , and Λ in our convention. Finally, we summarize the classification of mass matrices for neutrino models in Table 2.

Numerical results
We discuss numerical results for neutrino models in Table 2. The lepton mass matrices in the previous section are given by modulus parameter τ . By fixing τ , the modular invariance is broken, and then the lepton mass matrices give the mass eigenvalues and flavor mixing numerically. In order to fix the value of τ , we use the result of NuFIT 3.2 with the 3 σ error-bar [35]. We consider both the normal hierarchy (NH) of neutrino masses m 1 < m 2 < m 3 and the inverted hierarchy (IH) of neutrino masses m 3 < m 1 < m 2 , where m 1 , m 2 , and m 3 denote three light neutrino masses. The sum of neutrino masses are restricted by the cosmological observations [36,37]. Planck 2018 results provide us its cosmological upper bound for sum of neutrino masses; 120-160 meV [38] at the 95% C.L. depending on the combined data. We have used the upper bound of 160 meV as a conservative constraint of our models. By inputting the data of ∆m 2 atm ≡ m 2 3 − m 2 1 , ∆m 2 sol ≡ m 2 2 − m 2 1 , and three mixing angles θ 23 , θ 12 , and θ 13 with 3 σ error-bar given in Table  3, we fix the modulus τ and the other parameters. Then we can predict the CP violating Dirac phases δ CP and Majorana phases α 31 , α 21 , which are defined in Appendix C.

Model I(a): Seesaw
The coefficients α/γ and β/γ in the charged lepton mass matrix are given only in terms of τ after inputting the observed values m e /m τ and m µ /m τ as shown in Appendix B. Then, we have two free parameters, g 1 /g 2 and the modulus τ apart from the overall factors in the neutrino sector.
Since these are complex, we set The fundamental domain of τ is presented in Ref. [28]. In practice, we restrict our parametric search in Re[τ ] ∈ [−1.5, 1.5] and Im[τ ] > 0.6. We also take φ g ∈ [−π, π]. These four parameters are fixed by the observed ∆m 2 sol /∆m 2 atm and three mixing angles θ 23 , θ 12 and θ 13 .  At first, we present the prediction of the Dirac CP violating phase δ CP versus sin 2 θ 23 for NH of neutrino masses in Fig.1. It is emphasized that sin 2 θ 23 is restricted to be larger than 0.54, and δ CP = ±(50 • -180 • ). Since the correlation of sin 2 θ 23 and δ CP is characteristic, this prediction is testable in the future experiments of neutrinos. On the other hand, predicted sin 2 θ 12 and sin 2 θ 13 cover observed full region with 3 σ error-bar, and there are no correlations with δ CP .    Table 3 for model I(a). Results do not change under the exchange of α/γ and β/γ.
We also show the predicted Jarlskog invariant J CP [39], characterizing the magnitude of CP violation in neutrino oscillations, versus sin 2 θ 23 for NH of neutrino masses in Fig.2. The magnitude of J CP is predicted to be 0 -0.035 depending on θ 23 .
Let us show the prediction of the effective mass m ee which is the measure of the neutrinoless double beta decay as seen in Appendix C. The prediction of m ee is presented versus m 1 in Fig.4. It is remarkable that m ee is around 22 meV while m 1 is 40 meV. The red vertical line in Fig.4 denotes the upper bound of m 1 , which is derived from the cosmological bound m i < 160 meV. The obtained value of m 1 indicates near degenerate neutrino mass spectrum, m 1 m 2 40 meV and m 3 60 meV. The prediction of m ee 22 meV is testable in the future experiments of the neutrinoless double beta decay. We predict the rather large sum of neutrino masses as m i 145 meV, which is required by consistency with the observed value of sin 2 θ 13 .
The parameters of our model are determined by the input data of Table 3. Numerical values are listed in Table 4.
We have also scanned the parameter space for the case of IH of neutrino masses. We have found parameter sets which fit the data of ∆m 2 sol , ∆m 2 atm and three mixing angles sin 2 θ 23 , sin 2 θ 12 , and sin 2 θ 13 . However, the predicted m i is around 190-200 meV. Therefore, we also omit to show numerical results.

Model I(b): Seesaw
There is another assignment of the modular weight for the left-handed lepton and the righthanded charged leptons as presented in parentheses of Table 1 [31]. Then, the Dirac neutrino mass matrix is given by the constant parameter as seen in Eq. (22). We have scanned the parameter space for both NH and IH of neutrino masses. The parameters to reproduce the observed ∆m 2 sol and ∆m 2 atm cannot give the large mixing angle of θ 23 . The predicted value sin 2 θ 23 0.18 for NH. We also obtain sin 2 θ 12 0.8 and sin 2 θ 13 0.15. On the other hand, the predicted value sin 2 θ 23 0, sin 2 θ 12 0.5, and sin 2 θ 13 0 for IH. In conclusion, the model I(b) is inconsistent with the experimental data of Table 3.
It may be useful to add the discussion on the model by Criado and Feruglio [31], where the charged lepton mass matrix is different from ours in Eq. (18), but given by a flavon while the neutrino mass matrix is just same one in model I(b). We have reproduced the numerical results of Ref. [31], in which the three mixing angles and masses are consistent with the experimental data and the cosmological bound, respectively, for NH of neutrino masses. The predicted CP violating phase is δ CP ±100 • .

Model II: Weinberg Operator
In this case, the modulus τ is the only parameter in the neutrino mass matrix apart from the overall factors. We can find the parameter space to be consistent with the observed sin 2 θ 12 as well as ∆m 2 sol and ∆m 2 atm for both NH and IH. However, the predicted sin 2 θ 23 is around 0.8 and sin 2 θ 13 is very large as 0.45 for NH. On the other hand, for IH, the predicted sin 2 θ 23 is rather small as 0.35 and sin 2 θ 13 is around 0.04, which is larger than 1.6 times of the observed value. Thus, the neutrino mass matrix by the Weinberg operator do not lead to the realistic flavor mixing.

Model III: Dirac Neutrino
There is still a possibility of the neutrino being the Dirac particle. Then, the neutrino mass matrix is different from the Majorana one as shown in Table 2 although parameters are τ and g likewise in the case of the seesaw model I(a).
We have found the parameter space to be consistent with both observed sin 2 θ 23 and sin 2 θ 12 as well as ∆m 2 sol and ∆m 2 atm for NH. However, the predicted sin 2 θ 13 is much smaller than the observed value of O(10 −3 ).

Im[τ ]
Re  On the other hand, the sin 2 θ 13 is completely consistent with the observed value for IH of neutrino masses. We present the prediction of the Dirac CP violating phase δ CP versus sin 2 θ 23 for IH in Fig.5. The predicted δ CP is still allowed in [−π, π] depending on the magnitude of  We also show the predicted Jarlskog invariant J CP versus sin 2 θ 23 for IH of neutrino masses in Fig.6. The magnitude of J CP is predicted to be 0 -0.035.
The m i is required in 102 -150 meV to be consistent with the observed value of sin 2 θ 13 . We summarize numerical values of parameters in Table 5. In the Dirac neutrino model, the neutrinoless double beta decay is forbidden.

Summary
We study the phenomenological implications of the modular symmetry Γ(3) A 4 facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of the modulus τ , which is the only source of modular invariance breaking. We introduce no flavons in contrast with conventional flavor models with the A 4 symmetry. We classify the neutrino models along with type I seesaw (model I(a) and I(b)), Weinberg operator (model II), and Dirac neutrino (model III). For the charged lepton mass matrix, three right-handed charged leptons e R , µ R , and τ R are assigned to three different singlets 1, 1 , and 1 of A 4 , respectively.
For NH of neutrino masses, we have found that the seesaw model I(a) is available facing recent experimental data of NuFIT 3.2 [35] and the cosmological bound of the sum of neutrino masses [38]. The predicted sin 2 θ 23 is restricted to be larger than 0.54 and δ CP = ±(50 • -180 • ). The sharp correlation between sin 2 θ 23 and δ CP is testable in the future experiments of the neutrino oscillations. It is remarkable that m ee is around 22 meV while the sum of neutrino masses is 145 meV.
For IH of neutrino masses, the Dirac neutrino model III is completely consistent with the experimental data of NuFIT 3.2 and the cosmological bound of the sum of neutrino masses. The predicted δ CP is still allowed in [−π, π] depending on the magnitude of sin 2 θ 23 . The m i = 102 -150 meV is required by consistency with the observed value of sin 2 θ 13 .
The seesaw model I(b) and the Weinberg operator model II cannot reproduce the observed mixing angles after inputting the data of ∆m 2 sol and ∆m 2 atm for both NH and IH. It is helpful to comment on the effects of the supersymmetry (SUSY) breaking and the radiative corrections because we have discussed our model in the limit of exact SUSY. The SUSY breaking effect can be neglected if the separation between the SUSY breaking scale and the SUSY breaking mediator scale is sufficiently large [31]. In our numerical results, the corrections by the renormalization are very small as far as we take the relatively small value of tan β.
We have focused on the models with Yukawa couplings and masses, which correspond to modular forms of lower weights. Such models have strong constraints. Similarly, we can construct models with modular forms of higher weights. Such models would have a variety in model building.
lepton masses. We rewrite the mass matrix of Eq.(18) as whereα ≡ α/γ,β ≡ β/γ, We use the relation Y 2 2 + 2Y 1 Y 3 = 0 to eliminate Y 1 in the equation. Then, we obtain the following three equations: where χ ≡ m 2 e m 2 µ + m 2 µ m 2 τ + m 2 τ m 2 e . The coefficients C 1 , C 2 , and C 3 depend only onŶ 2 ≡ Y e iφ Y , where Y is real positive and φ Y is a phase parameter, These values are determined if the value of modulus τ is fixed. Then, we obtain the general equations which describeα andβ as functions of charged lepton masses and τ : where we redefine the parametersα 2 +β 2 = s andα 2β2 = t. They are related as follows,

C Lepton mixing and neutrinoless double beta decay
Supposing neutrinos to be Majorana particles, the PMNS matrix U PMNS [40,41] is parametrized in terms of the three mixing angles θ ij (i, j = 1, 2, 3; i < j), one CP violating Dirac phase δ CP , and two Majorana phases α 21 , α 31 as follows: where c ij and s ij denote cos θ ij and sin θ ij , respectively.
The rephasing invariant CP violating measure, the Jarlskog invariant [39], is defined by the PMNS matrix elements U αi . It is written in terms of the mixing angles and the CP violating Dirac phase as: J CP = Im U e1 U µ2 U * e2 U * µ1 = s 23 c 23 s 12 c 12 s 13 c 2 13 sin δ CP .